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Réalisations des espaces de Besov homogènes. (Realization of homogeneous Besov spaces). (French) Zbl 0661.46026
The homogeneous Sobolev space $$\dot H^ 1({\mathbb{R}}^ n)$$ is the space of locally integrable functions f such that $$\nabla f\in L^ 2({\mathbb{R}}^ n)$$. Thus, $$f\in L^ 2({\mathbb{R}}^ n)$$ is not required. The main advantage is that the $$L^ 2$$ norm of $$\nabla f$$ is invariant under dilations. But, in order to obtain a Banach space for the natural norm, we need to reinterpret the elements of this and other homogeneous spaces as equivalence classes (modulus polynomials). Hence, we are led to the problem of representing or realizing these spaces. This means to select a unique element of each equivalence class in a linear and continuous way. Then the question is whether these realizations preserve translation and dilation invariance. In this respect the paper includes the following two theorems, where $$\dot H^ s_ p$$ and $$\dot B_ p^{s,q}$$ stand for the homogeneous Sobolev and Besov spaces on $${\mathbb{R}}^ n$$, respectively. (These spaces are homogeneous of degree s-n/p).
Theorem 1. (i) If $$s<n/p$$ and $$q<\infty$$ the spaces $$\dot H^ s_ p$$ and $$\dot B_ p^{s,q}$$ have a unique translation invariant realization. This also holds for $$\dot B_ p^{n/p,1}.$$
(ii) If $$s>n/p$$ the spaces $$\dot H^ s_ p$$ and $$\dot B{}_ i^{s,q}$$ have no translation invariant realization. This also holds for $$\dot B{}_ p^{n/p,q}(q>1)$$ and $$\dot H_ p^{n/p}.$$
Theorem 2. (i) If $$s-n/p\not\in N$$ the spaces $$\dot H^ s_ p$$ and $$\dot B{}_ p^{s,q}$$ have a unique dilation invariant realization.
(ii) If $$s-n/p\in N$$ and $$q>1$$ the space $$\dot B{}_ p^{s,q}$$ has no dilation invariant realization. This also holds for $$\dot H^ s_ p.$$
(iii) The space $$\dot B_ p^{s,1}$$ with $$s-n/p\in N$$ has infinitely many dilation invariant realizations.
Reviewer: F.Bernis

MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:
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